# Thread: Accumulation points and finding them

1. ## Accumulation points and finding them

This is the definition of accumulation point that my book gives:

A is an accumulation point of $S \subset \mathbb{R}, \forall \epsilon > 0, S \bigcap B(A;\epsilon)$ is infinite.

The book I have gives horrible examples on what accumulation points actually are (contradicting itself two out of the three times), but never actually gives instructions on how to find the points.

This is the question I have to solve:

Find the accumulation points of $S = \left\{\frac{2}{n} + (1 - \frac{1}{n})cos(\frac {n\pi}{2}) : n \in\mathbb{N}\right\}$

Can anyone help to actually explain to me, in english, what an accumulation point is? I tried Wikipedia but it's more of this meaningless jargon.

Hopefully, understanding what it is I'm looking for will show me how to answer this question. If not, I could use help there too.

2. This is what I've come to thus far on a different forum:

It looks like it will just go back to looking like a typical cosine graph (which I guess is to be expected since 2/n and 1/n go to 0)

So if we set $n_{k} = 2k \Rightarrow\left\{\frac{1}{k} + (1 - \frac{1}{2k})cos(k\pi) \right\} \rightarrow (-1,1)$

and $n_{k+1} = 2k+1 \Rightarrow\left\{\frac{2}{2k+1} + (1 - \frac{1}{2k+1})cos(\frac{\pi(2k+1)}{2}) \right\} \rightarrow {0}$

Ok I think I broke my brain. Did any of that make sense? If it does, are the accumulation points for the set -1,0,1?

If it didn't make any sense (because I'm supposing it doesn't at this point, I've had a bit of a head cold all weekend), where should I actually be heading?

3. *delete*